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In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.
Sir William Vallance Douglas Hodge FRS FRSE [2] (/ h ɒ dʒ /; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer. [ 3 ] [ 4 ] His discovery of far-reaching topological relations between algebraic geometry and differential geometry —an area now called Hodge theory and pertaining more generally to Kähler ...
Hodge theory, named after W. V. D. Hodge, is originally formulated for the de Rham complex. Practically ,it can be used to study Riemannian and Kahler manifold and algebraic geometry of complex projective varieties . It is related to the study of nonlinear PDEs.
For this paperback edition, Professor Sir Michael Atiyah has written a foreword that sets Hodges work in its historical context and relates it briefly to developments. First published in 1941,...
(1) 4 is of the second kind, if given any point P on Vm, there exists a mero-morphic (q - 1)-form w, defined globally on Vm , such that 4 - dw is holomorphic in the neighborhood of P; (2) 4. is of the second kind if l = 0f frO where r is any q-cycle of Vm - W which bounds in Vm. Picard and Lefschetz both prove the equivalence of these ...
The w.v. d. Hodge provides a way to analyze the relationships between different types of cohomology classes on smooth projective varieties. It emphasizes the significance of vertical forms and their contribution to the overall structure of differential forms in Hodge theory.
William Hodge was Lowndean Professor of Astronomy and Geometry at Cambridge. His main interests were in Algebraic Geometry and Differential Geometry.
